Maximin share and minimax envy in fair-division problems

For fair-division or cake-cutting problems with value functions which are normalized positive measures (i.e., the values are probability measures) maximin-share and minimax-envy inequalities are derived for both continuous and discrete measures. The tools used include classical and recent basic convexity results, as well as ad hoc constructions. Examples are given to show that the envy-minimizing criterion is not Pareto optimal, even if the values are mutually absolutely continuous. In the discrete measure case, sufficient conditions are obtained to guarantee the existence of envy-free partitions.     

Homogeneous quaternionic Kähler structures of linear type

A classification of homogeneous quaternionic Kähler structures by real tensors is given and related to Fino's representation theoretic decomposition. A relationship between the modules whose dimension grows linearly and quaternionic hyperbolic space is found. To cite this article: M. Castrillón López et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A Note on Cactus Trees: Variational vs. Recursive Approach

In this article we deal with the variational approach to cactus trees (Husimi trees) and the more common recursive approach, that are in principle equivalent for finite systems. We discuss in detail the conditions under which the two methods are equivalent also in the analysis of infinite (self-similar) cactus trees, usually investigated to the purpose of approximating ordinary lattice systems. Such issue is hardly ever considered in the literature. We show (on significant test models) that the phase diagram and the thermodynamic quantities computed by the variational method, when they deviates from the exact bulk properties of the cactus system, generally provide a better approximation to the behavior of a corresponding ordinary system. Generalizing a property proved by Kikuchi, we also show that the numerical algorithm usually employed to perform the free energy minimization in the variational approach is always convergent.     

1/N-Expansion for the Dicke model and the decoherence program

An analysis of the Dicke model, N two-level atoms interacting with a single radiation mode, is done using the Holstein-Primakoff transformation. The main aim of the paper is to show that, changing the quantization axis with respect to the common usage, it is possible to prove a general result either for N or the coupling constant going to infinity for the exact solution of the model. This completes the analysis, known in the current literature, with respect to the same model in the limit of N and volume going to infinity, keeping the density constant. For the latter the proper axis of quantization is given by the Hamiltonian of the two-level atoms and for the former the proper axis of quantization is defined by the interaction. The relevance of this result relies on the observation that a general measurement apparatus acts using electromagnetic interaction and so, one can state that the thermodynamic limit is enough to grant the appearance of classical effects. Indeed, recent experimental results give first evidence that superposition states disappear interacting with an electromagnetic field having a large number of photons.     

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i _X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i _X −1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i _X ≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρ_X> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.     

An experimental analysis of a population based approach for global optimization

In this paper we perform a computational analysis of a population based approach for global optimization, Population Basin Hopping (PBH), which was proven to be very efficient on very challenging global optimization problems by the authors (see ). The experimental analysis aims at understanding more deeply how the approach works and why it is successful on challenging problems.     

Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow

A time-dependent model corresponding to an Oldroyd-B viscoelastic fluid is considered, the convective terms being disregarded. Global existence in time is proved in Banach spaces provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes problem. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates, using again an implicit function theorem. Supported by the Swiss National Science Foundation. Fellowship PBEL2–114311.     

How to assess and report the performance of a stochastic algorithm on a benchmark problem: mean or best result on a number of runs?

Some authors claim that reporting the best result obtained by a stochastic algorithm in a number of runs is more meaningful than reporting some central statistic. In this short note, we analyze and refute the main argument brought in favor of this statement.